Malika travelled `5/16` of his journey by coach and 7/20 by rail, she walked the remaining 10 km, how far did she go altogether?

Options

A. `27 17/29`
B. `33 7/27`
C. `19 7/27`
D. `29 17/27`

Answer

To find the total distance that Malika traveled, we can follow these steps:

### Step 1: Define the total distance
Let the total distance Malika traveled be represented as \( D \).

### Step 2: Calculate the distance traveled by coach
Malika traveled \( \frac{5}{16} \) of her journey by coach. Therefore, the distance traveled by coach is:
\[ \text{Distance by coach} = \frac{5}{16} D \]

### Step 3: Calculate the distance traveled by rail
Malika traveled \( \frac{7}{20} \) of her journey by rail. Therefore, the distance traveled by rail is:
\[ \text{Distance by rail} = \frac{7}{20} D \]

### Step 4: Calculate the distance walked
Malika walked the remaining distance, which is given as 10 km.

### Step 5: Set up the equation
The total distance can be expressed as the sum of the distances traveled by coach, rail, and walking:
\[ \frac{5}{16} D + \frac{7}{20} D + 10 = D \]

### Step 6: Find a common denominator
To solve this equation, we need to find a common denominator for the fractions \( \frac{5}{16} \) and \( \frac{7}{20} \). The least common multiple (LCM) of 16 and 20 is 80.

### Step 7: Rewrite the fractions
Now we can rewrite the fractions:
\[ \frac{5}{16} = \frac{5 \times 5}{16 \times 5} = \frac{25}{80} \]
\[ \frac{7}{20} = \frac{7 \times 4}{20 \times 4} = \frac{28}{80} \]

### Step 8: Substitute back into the equation
Substituting these values into the equation gives:
\[ \frac{25}{80} D + \frac{28}{80} D + 10 = D \]

### Step 9: Combine the fractions
Combining the fractions on the left side:
\[ \frac{25 + 28}{80} D + 10 = D \]
\[ \frac{53}{80} D + 10 = D \]

### Step 10: Isolate \( D \)
Now, we can isolate \( D \):
\[ 10 = D - \frac{53}{80} D \]
\[ 10 = \left(1 - \frac{53}{80}\right) D \]
\[ 10 = \frac{80 - 53}{80} D \]
\[ 10 = \frac{27}{80} D \]

### Step 11: Solve for \( D \)
To find \( D \), multiply both sides by \( \frac{80}{27} \):
\[ D = 10 \times \frac{80}{27} \]
\[ D = \frac{800}{27} \approx 29.63 \text{ km} \]

### Conclusion
Thus, the total distance Malika traveled is approximately \( 29.63 \) km.

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